I think that analyzing musical scores might mean different things to different people, depending on the intent of the analysis and the degree of detail involved. An understanding of
musical form is basic to recognizing music's structural elements, and fairly solid grasp of
music theory is needed to examine the devices employed within a composition.
If you're studying a typical sonata, for example, it might be useful to be familiar with sonata-allegro form (and to identify the exposition, development and recapitulation sections that characterize it). Many musicians find that studying patterns is beneficial to memorization; they may be linear (e.g., intervals comprising melodies), vertical (chords and voicings) or both (harmonic progressions and cadences). And plenty of people playing classical music get by without formal knowledge of any of those things; their knowledge of “theory” may extend to knowing the key signature of a given piece, if that.
Regarding the other question of whether it makes most sense to learn music from various eras or to concentrate on one in particular, it’s commonly advised that studying pieces representing a broad range of periods and styles is fundamental to well-rounded musicianship. I believe that’s very sound advice for the serious student, but it’s nevertheless common for professionals to specialize in a genre they find most congenial.
The adult amateur is also likely to gravitate to a favorite era or a particular composer; considering the importance of maintaining motivation in the pursuit of musical practice for one’s own pleasure, I think that makes sense. I believe in keeping an open mind (and open ears!), but I only learn music that I love.